Videos
Donkey Kong Part 3 - Did You Know Gaming? Feat. TheCartoonGamer
 
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If the Player backtracks at a certain point in level 4-3, The Amiss Abyss level, and waits a while they can see a one of the eponymous creatures of the Metroid Franchise floating in the background in silhouette.

This is true in both the Wii U and Nintendo Switch versions of the game
Contributed by PirateGoofy
A string of text labelled "miiverse", found in the game files, suggests there was originally going to be an auto-post achievement system for when the player did certain things, similar to Hyrule Warriors.
Contributed by Boyobmas
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At the end of the level, Aquedect Assault, the arcade sprite of the Original Donkey Kong can be seen in the background, much like Wonky Waterway from Donkey Kong Country Returns, only holding a Wii U Gamepad instead of a Wii Remote.
Contributed by CuriousUserX90
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The Giant Squeekly from Crowded Cavern in Donkey Kong Country Returns can be seen frozen solid in the Background of the level, Blurry Furry.
Contributed by CuriousUserX90
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At the second checkpoint in Fruity Factory, the background features a replica of Stage 1 from the original Arcade/NES classic, Donkey Kong 3.
Contributed by Gamer#1
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Samus' ship from the Metroid series can be seen in the background of the Busted Bayou level.
Contributed by Boyobmas
At the beginning of level 1-1, Mangrove Cove, if the player turns around and enters the crashed plane, there will be a pile of four TVs. If Donkey Kong uses his ground pound move while near the four TVs, they will turn on. Two of the TVs will have the title screen for the English version of Donkey Kong Country: Returns playing, while the other two will have the Japanese version playing.
Contributed by Charlie
One of the items in Funky Kong's shop is a blue shield called the Crash Guard. His commentary on the item is a reference to the original Legend of Zelda: "Vehicle troubles? It's dangerous to go alone--take one of these!"
Contributed by JiraiyaSennin
There's an old-fashioned record player in a hidden area at the end of World 1-3, near the secret exit. It strongly resembles Cranky Kong's record player seen in the original Donkey Kong Country, and will even play the same tune if the player pounds the ground next to it.
Contributed by Boyobmas
If DK is left idle for long enough, he will start playing a blue Nintendo 3DS XL, similar to to how he would play a grey Nintendo DS in DKC Returns. The audio heard from the system is actually from four real 3DS games by Nintendo:
Animal Crossing: New Leaf
Mario Kart 7
Super Mario 3D Land
Donkey Kong Country Returns 3D
Contributed by Boyobmas